The present invention relates to fiber optic gyroscopes used for rotation sensing and, more particularly, to interferometric fiber optic gyroscopes.
Fiber optic gyroscopes are an attractive means with which to sense rotation. They can be made quite small and still be constructed to withstand considerable mechanical shock, temperature change, and other environmental extremes. In the absence of moving parts, they can be nearly maintenance free, and they have the potential to become economical in cost. They can also be sensitive to low rotation rates which can be a problem in other types of optical gyroscopes.
There are various forms of optical inertial rotation sensors which use the well-known Sagnac effect to detect rotation about a pertinent axis thereof. These include active optical gyroscopes which have the gain medium contained in an optical cavity therein, such as the ring laser gyroscope, and passive optical gyroscopes without any gain medium in the primary optical path, such as the interferometric fiber optic gyroscope and the ring resonator fiber optic gyroscope. The avoidance of having the active medium along the primary optical path in the gyroscope eliminates some problems which are encountered in active gyroscopes, such as low rotation rate lock-in, bias drift and some causes of scale factor variations.
Interferometric fiber optic gyroscopes typically employ a single spatial mode optical fiber of a substantial length, typically 100 to 2,000 meters, which length is formed into a coil by being wound on a core to form a closed optical path. An electromagnetic wave, or light wave, is introduced and split into a pair of such waves to propagate in opposite directions through the coil to both ultimately impinge on a photodetector. Rotation about the sensing axis of the core, or the coiled optical fiber, provides an effective optical path length increase in one rotational direction and an effective optical path length decrease in the opposite rotational direction for one member of this pair of electromagnetic waves. The opposite result occurs for the remaining member of the pair of electromagnetic waves for such rotation. Such path length differences between the pair of electromagnetic waves introduce a phase shift between those waves in interferometric fiber optic gyroscopes in either rotation direction, i.e. the well-known Sagnac effect. The use of a coiled optical fiber is desirable because the amount of phase difference shift due to rotation, and so the output signal, depends on the length of the entire optical path through the coil traversed by the two opposing directional electromagnetic waves. Thus, a relatively large phase shift difference can be obtained in a long optical fiber, but also in the relatively small volume taken by that fiber in its being coiled.
The output current from the photodetector system photodiode in response to the opposite direction traveling electromagnetic waves impinging thereon, after passing through the coiled optical fiber, follows a raised cosine function, that is, the output current depends on the cosine of the phase difference between these two waves. Since a cosine function is an even function, such an output function gives no indication as to the relative direction of the phase difference shift, and so no indication as to the direction of the rotation about the axis. In addition, the rate of change of a cosine function near zero phase value is very small, and so such an output function provides very low sensitivity for low rotation rates.
Because of these unsatisfactory characteristics, the phase difference between the two electromagnetic waves is usually modulated by placing an optical phase modulator on one side of the coiled optical fiber. As a result, one of the opposite direction propagating waves passes through the modulator just after entering the coil, while the other wave, traversing the coil in the opposite direction, passes through the modulator just before exiting the coil. In addition, a phase sensitive demodulator is provided to receive the photodetector output current. Both the optical phase modulator and the phase sensitive demodulator are typically operated by a sinusoidal signal generator, but other waveform types of a similar fundamental frequency can also be used.
The resulting signal output of the phase sensitive demodulator follows a sine function, i.e. the output signal depends on the sine of the phase difference between the two electromagnetic waves impinging on the photodiode, primarily the phase shift due to rotation about the axis of the coil. A sine function is an odd function having its maximum rate of change at zero, and so changes algebraic sign on either side of zero. Hence, the phase sensitive demodulator signal can provide both an indication of which direction a rotation is occurring about the axis of the coil, and the maximum rate of change of signal value as a function of rotation rate near a zero rotation rate. That is, the signal has its maximum sensitivity near zero phase shift so that its output signal is quite sensitive to low rotation rates. This is possible, of course, only if phase shifts due to other sources, that is, errors, are made sufficiently small. In addition, this output signal in these circumstances is very close to being linear at relatively low rotation rates. Such characteristics for the output signal of the phase sensitive demodulator is a substantial improvement over the characteristics of the output current of the photodetector.
Reducing erroneous phase shifts from other sources is, however, a difficult problem in fiber optic gyroscopes. Avoidance of erroneous phase shifts in the electromagnetic waves reaching the photodetector requires that each of the interfering waves, at least those of the same wavelength, have traveled over the same optical path, that is, the electromagnetic wave of a wavelength associated with a clockwise direction of travel from the coil and the one of the same wavelengths associated with the counterclockwise direction of the coil each must travel over an indistinguishable optical path from the source to the photodetector absent any rotation of the coil. A system with this characteristic is often termed "reciprocal." At a minimum, the optical paths corresponding to the common wavelength clockwise electromagnetic waves and counterclockwise electromagnetic waves must be identical on an optical ray tracing basis in the absence of rotation. In meeting this requirement, a "minimum reciprocal configuration" has been found to be as shown in FIG. 1 in connection with the coiled optical fiber, 10, shown there. Coiled optical fiber 10 in FIG. 1 is, as indicated above, wound about a core or spool using primarily an ordinary single spatial mode optical fiber wrapped about an axis thereof which becomes the axis about which rotation is to be sensed. The use of such a single mode fiber allows the paths of the electromagnetic waves to be defined nearly uniquely, and further allows the phase fronts of such a guided wave to be defined uniquely. This greatly aids in maintaining reciprocity.
However, the optical fiber in coil 10 is not entirely ordinary single spatial mode optical fiber because of a depolarizer included relatively near one end thereof, although this depolarizer could be located anywhere in coil 10. The ordinary single spatial mode optical fiber used in a very great fraction of coil 10 is subject to having changing birefringence therein introduced by mechanical stress changing with temperature, and by the Faraday effect in magnetic fields. This changing birefringence will lead to randomly varying polarization rotations of the beams passing therethrough even to the extent of being so great that the interference of those beams at the photodetector vanishes. One method for solving this problem is the use of so-called polarization-maintaining optical fiber exhibiting a high birefringence in place of the ordinary single spatial mode fiber without such marked birefringence. The significant birefringence constructed into such a fiber leaves birefringence changes from other sources being relatively insignificant.
However, such polarization-maintaining optical fiber is relatively expensive so that there is a substantial desire to be able to use just ordinary single spatial mode optical fiber. That desire can be satisfied with the use of a depolarizer, 10', located within coil 10 and shown in FIG. 1 to be positioned near one end in coil 10 for ease of winding that coil. Such a depolarizer tends to closely equalize the electromagnetic wave intensities in, and decorrelate, the two orthogonal polarization modes permitted therein and overwhelm the effects of the randomly changing birefringence in the ordinary single spatial mode fiber in the rest of coil 10 thus preventing such opposing direction beam interference at the optical subsystem output photodiode from vanishing.
Such a depolarizer can be formed with two lengths of polarization-maintaining fiber, 10" and 10'", with the latter being substantially twice as long as the former to thereby cause approximately twice the optical delay caused by the other. In each of these lengths, there is a high refractive index axis, i.e. the slower propagation axis or the "x" axis, and a low refractive index axis, i.e. the faster propagation axis or "y" axis, which are orthogonal to one another. The lengths are joined in a fused splice in such a manner that the "x" axis in one length is approximately equidistant from the "x" and "y" axes in the other length, i.e. the "x" axis in the former is at 45.degree. from each of the "x" and "y" axes in the other. The opposite ends of each of the depolarization fiber lengths are then spliced by fusing to corresponding portions of the single spatial mode ordinary optical fiber in coil 10 so that a beam of light, propagating through any of the depolarizer or either of the single spatial mode ordinary optical fiber portions, substantially propagates through all of them.
Coil 10 with depolarizer 10' is typically wound on a spool using the "quadripole" technique so that similarly located points in the coil with respect to center are near one another. This reduces the effects of time-varying phenomena, such as thermal gradients, from affecting opposite direction propagating electromagnetic waves differently from one another.
The electromagnetic waves which propagate in opposite directions through coil 10 are provided from an electromagnetic wave source, or light source, 11, in FIG. 1. This source is typically a superluminescent diode or, alternatively, a laser diode operating below its threshold for stimulated emission, either of which provide electromagnetic waves typically in the near-infrared part of the spectrum with a typical wavelength of 830 nm. Source 11 must have a short coherence length for emitted light to reduce the phase shift difference errors between these waves due to Rayleigh scattering at scattering sites in coil 10. Because of the nonlinear Kerr effect in coil 10, differing intensities in the two propagating waves can also lead to phase difference shifts therebetween. This situation can also be aided by the use of a short coherence length source for source 11 which leads to modal phase shift canceling. Rayleigh scattering and the nonlinear Kerr effect lead to non-reciprocal phase shifts between the counter-rotating electromagnetic waves in coil 10 even in a minimum reciprocal configuration. A superluminescent diode, or a laser diode operating below threshold, each have a wide emission spectrum compared to that of a laser diode operating past its threshold in the stimulated emission mode of operation.
Between laser diode 11 and fiber optic coil 10 in FIG. 1 there is shown an optical path arrangement formed by an extension of the ends of the optical fiber forming coil 10 to some optical coupling components which separate the overall optical path into several optical path portions. A portion of polarization-maintaining optical fiber is positioned against a face of laser diode 11 at a location of optimum light emission therefrom, a point from which it extends to a first optical directional coupler, 12, to be joined thereto. If, on the other hand, coupler 12 is formed by fusing two optical fibers together in a coupling region, either a pair of polarization-maintaining optical fibers or a pair of ordinary single spatial mode optical fibers, the excess length of one of the optical fibers may be positioned against diode 11 to provide this optical path between diode 11 and this wave coupling region of coupler 12, or the excess length may be spliced to another polarization-maintaining or ordinary single mode optical fiber, respectively, extending from diode 11.
Optical directional coupler 12 has light transmission media therein which extend between four ports, two on each end of that media, and which are provided on each end of coupler 12 in FIG. 1. One of these ports has the optical fiber extending from laser diode 11 positioned thereagainst (or vice versa for a fused coupler, i.e. a fiber extending from the coupler coupling region to be positioned against the emitting face of diode 11). At the other port on the same end of optical coupler 12 there is shown a further optical fiber positioned thereagainst (or alternatively extending from the fused coupler if used) which extends to be positioned against a photodiode, 13, which is electrically connected to a photodetection system, 14. This optical fiber may be a polarization-maintaining optical fiber or it may be an ordinary single spatial mode optical fiber. In practice, as indicated above, coupler 12 may be formed from fused lengths of such optical fiber so that the remaining lengths past the fused portion, or the light coupling region therein, extend either all the way to laser diode 11 and photodiode 13, or are spliced to other optical fibers extending therefrom.
Photodiode 13 detects electromagnetic waves, or light waves, impinging thereon from the portion of the optical fiber positioned thereagainst (or extending thereto) and provides a photocurrent in response. This photocurrent, as indicated above, in the situation of two nearly coherent electromagnetic waves impinging thereon, follows a raised cosine function in providing a photocurrent output which depends on the cosine of the phase difference between such a pair of electromagnetic waves. Photodiode 13 is operated in either the photovoltaic mode or the photoconductive mode, as needed, into an amplifier circuit of appropriate impedance to provide a photocurrent which is substantially a linear function of the impinging radiation intensity, and is typically a p-i-n photodiode.
Optical directional coupler 12 has another polarization-maintaining optical fiber against a port at the other end thereof which extends to a polarizer, 15. Again, the excess length in an optical fiber past the coupling region in coupler 12 may have the end thereof extend all the way to polarizer 15, or may be spliced to another optical fiber portion extending from polarizer 15 with the principal birefringence axes in each such portion of optical fiber closely aligned to those of the other. At the other port on that same side of coupler 12 there is a non-reflective termination arrangement, 16, involving the excess length of one of the optical fibers fused together forming coupler 12 or, again, another optical fiber spliced to such an excess length. This optical fiber leading to arrangement 16 can again be polarization-maintaining optical fiber or ordinary single spatial mode optical fiber.
Directional optical coupler 12, in receiving electromagnetic waves, or light, at any port, or at any end of an excess portion of optical fiber extending past the coupling region therein, transmits such electromagnetic waves so that a preselected fraction thereof, typically one-half, appears at each of the two ports, or ends of the two excess optical fiber lengths past the coupling region, which are at the opposite end of coupler 12 from that having the incoming port or excess optical fiber length receiving the incoming waves. On the other hand, no electromagnetic waves are transmitted to the port or excess fiber length which is on the same end of coupler 12 as is the incoming port. The polarization of the incoming electromagnetic waves with respect to the principal refringent axes at the input port can be fairly well preserved at the corresponding axes of the two output ports if coupler 12 is formed of two portions of polarization-maintaining optical fiber with the principal axes suitably aligned, but there will be some coupling of waves between axes in the coupling region of the coupler. If a pair of ordinary single spatial mode optical fiber portions are fused together to form coupler 12, the polarization of the incoming electromagnetic waves with respect to the principal birefringent axes in one component fiber can be fairly well preserved through the coupling region to the other fiber, but there may be substantial coupling thereafter even before coupled waves reach the output port of the ordinary single spatial mode optical fiber.
Polarizer 15 is used because, even in a single spatial mode optical fiber, two polarization modes are possible for electromagnetic waves passing through such a fiber along orthogonal axes. Thus, polarizer 15 is provided for the purpose of transmitting the electromagnetic wave component along one of these axes, for one of these polarization modes, between the optical fibers connected to the ports on either end thereof, i.e. between the "x" slow axis of the polarization-maintaining optical fiber connected thereto to provide a propagation path to and from directional coupler 12 and the ordinary single spatial mode optical fiber placed against the port on the opposite side thereof to be described below. At the same time, polarizer 15 substantially blocks transmission from the "y" or fast axis of the polarization-maintaining optical fiber extending between it and directional coupler 12 and the ordinary single spatial mode fiber on the opposite side thereof. Hence, the slow axis of the polarization-maintaining optical fiber extending from coupler 12 is aligned with the transmission axis of polarizer 15 at a port on one side thereof, or with the slow principal birefringent axis of an optical fiber connection portion extending from polarizer 15 that is closely aligned with the transmission axis of the polarizer. The fast axis of this optical fiber is then closely aligned to the blocking axis at the port of the polarizer, or to the fast principal birefringent axis of a connection optical fiber extending therefrom which is closely aligned with a polarizer blocking axis.
Polarizer 15, however, is not capable of entirely blocking electromagnetic waves in the one state of polarization that it is intended to block. This shortcoming in the extinction coefficient thereof leads to a non-reciprocity between two opposite direction traveling waves over the optical paths they follow, and so to a non-reciprocal phase shift occurring between them which can vary with the conditions of the environment in which the polarizer and the remainder of the system of placed.
Positioned against the port of polarizer 15 on the end opposite that connected with optical directional coupler 12, or spliced to a polarization-maintaining optical fiber length extending therefrom, is an ordinary single spatial mode optical fiber which extends to a further optical directional coupler, 17, a coupler which is typically formed of two portions of such ordinary single spatial mode fiber. Directional coupler 17 also transmits received electromagnetic waves so that a preselected fraction thereof, again typically one-half, appears at each of the two ports, or the ends of the two excess optical fiber lengths past the coupling region, which are at the opposite end of coupler 17 from that having the incoming port or having the excess optical fiber length receiving the incoming waves. Again, no electromagnetic waves are transmitted to the port or excess fiber length which is on the same end of coupler 17 as the incoming port. The polarization of incoming electromagnetic waves at an input port will not be very well preserved at the corresponding pair of output ports. Alternatively, directional coupler 17 could be formed using a pair of portions of polarization-maintaining optical fiber, but this will lead to somewhat different optical performance in the optical subsystem portion of FIG. 1 which would be similar to the performance of such a subsystem if directional coupler 17 was alternatively formed in an integrated optic chip.
If directional coupler 17 is formed by fusing together two optical fibers, the excess portion of one of the optical fibers therein past the coupler region therein may extend all the way to the appropriate port on one end of polarizer 15. In an alternative, this excess portion may be spliced to an ordinary single spatial mode optical fiber portion extending from polarizer 15 (or to a polarization-maintaining fiber portion extending therefrom).
The second port on the same end of coupler 17 from which the first port is coupled to polarizer 15 is connected in a non-reflective termination arrangement, 18, using a further ordinary single spatial mode optical fiber portion or the excess length of an optical fiber associated with that port beyond the coupling region in coupler 17 in the situation of fused optical fibers. One of the ports on the opposite end of coupler 17 is connected to a further optical component in the optical path portion extending thereto from one end of the optical fiber in coil 10. The other port on that end of coupler 17 is directly coupled to the remaining end of optical fiber coil 10 near which depolarizer 10' is located, and this coupling is typically accomplished through a splice between the excess length of an optical fiber past the coupling region in coupler 17 and the optical fiber in coil 10.
Between coil 10 and coupler 17, on the side of coil 10 opposite the directly connected side thereof, there is provided an optical phase modulator, 19. Optical phase modulator 19 has a port on either end of the transmission media contained therein which occur in FIG. 1 at the opposite ends of that phase modulator. The ordinary single spatial mode optical fiber from coil 10 is positioned against a port of modulator 19. The ordinary single spatial mode optical fiber extending from coupler 17 is positioned against the port on the opposite end of modulator 19.
Optical phase modulator 19 can be of the variety formed by wrapping an optical fiber portion around a piezoelectric cylinder so that the fiber may be stretched by the application of voltage to that cylinder, or this phase modulator may be formed as an optical integrated chip using a substrate of lithium niobate, for instance, with metallic depositions provided thereon as electrodes and positioned adjacent a waveguide provided therein. Such depositions typically result in plate-like electrode structures on the substrate to both provide electrical contacts to the modulator and a means through which varying electric fields can be established in the waveguide to result in the necessary modulation of the phase of electromagnetic waves passing through that waveguide.
Optical phase modulator 19 is thus capable of receiving electrical signals on these plates to cause the introduction of phase differences in electromagnetic waves transmitted therethrough by changing the index of refraction of the transmission medium, or transmission media, because of the resulting electric fields established therein to thereby change the effect of optical path lengths experienced by such waves. Optical phase modulators constructed in optical integrated circuit form have a large bandwidth, i.e. are able to provide phase changes following a waveform that has substantial high frequency content. Note also that polarizer 15, and source and loop optical directional couplers 12 and 17, could also be formed in similar integrated optic chips, including possibly being formed in a common such chip.
Directional optical coupler 17 serves as a beam-splitting apparatus in that electromagnetic waves emitted from source 11 that are transmitted through coupler 12 and polarizer 15 to be received by coupler 17 are there split in approximately half with a corresponding one of the resulting portions passing out of each of the two ports on the opposite end of coupler 17. Out of one port on that opposite end of coupler 17 the corresponding electromagnetic wave portion passes through depolarizer 10', the rest of optical fiber coil 10, through optical phase modulator 19 and back to coupler 17. A portion of that electromagnetic wave passes through the port of coupler 17 leading to polarizer 15 and then to coupler 12 where a part of the remainder of the wave portion is transmitted to photodiode 13.
The other portion of the electromagnetic wave after the split in coupler 17 leaves that other port on the coil 10 end of coupler 17 to first pass through optical phase modulator 19, through most of optical fiber coil 10, and then through depolarizer 10' to reenter coupler 17 and, again, from there follow the same path as the first portion previously described to finally impinge in part on photodiode 13. In the presence of modulation provided by phase modulator 19, and in the presence of any rotation of coil 10 about its axis, or because of effects in coupler 17, some of the energy of the combined waves will be lost through non-reflective arrangement 18.
In an interferometric fiber optic gyroscope using polarization-maintaining optical fiber for coil 10 without a depolarizer, the electromagnetic waves passing through coil 10 are all intended to take the same optical path. In the system of FIG. 1, however, the nature of the ordinary single spatial mode optical fiber used in coil 10 gives rise to random occurrences of birefringence therein induced by various causes, including stress change due to temperature changes, which result in the possibility of different optical paths being available for the waves to propagate over. Depolarizer 10' forces waves to differing polarization states periodically over wavelength, and so to corresponding different optical paths. Thus, the polarization history of electromagnetic waves through coil 10 and depolarizer 10' together is wavelength dependent. Nevertheless, any waves reaching the transmission axis of polarizer 15 at a point in time will have had the same polarization history. Assuming then that depolarizer 10' distributes the optical waves between the polarization states uniformly, depolarizer 10' acts to equalize the wave energy in each of the optical paths.
As indicated above, photodiode 13 provides an output current proportional to the intensity of the combined electromagnetic waves, or light waves, impinging thereon dependent on the phase difference therebetween. The arrangement of FIG. 1 leads to the electromagnetic waves propagating in opposite directions through coil 10 over various optical paths to in part reach photodiode 13 so that the intensity thereon is an average of the electromagnetic waves traveling in both directions over each polarization determined optical path, i.e. averaged over the wavelengths present, but including primarily only those waves propagating over those optical paths over which returning waves have a polarization at polarizer 15 which is substantially passed by that polarizer. That is, the returning waves included in the averaging process are primarily just those following optical paths which extend through the transmission axis of polarizer 15. Corresponding photocurrent from photodiode 13 follows a raised cosine function in being based on the cosine of the average phase difference between portions of each of the electromagnetic waves propagating in opposite directions in coil 10 impinging thereon taken over the wavelengths present therein. This relationship follows because the photocurrent depends on the resulting optical intensity of the pairs of opposite direction propagating electromagnetic waves incident on photodiode 13 which intensity will vary depending on how much constructive or destructive interference occurs between these waves at that diode. This interference of waves will change with rotation of the coiled optical fiber forming coil 10 about its axis as such rotation introduces a phase difference shift between the waves because of the Sagnac effect. Further, additional phase difference shifts will be introduced by optical phase modulator 19 as will be described in connection with the electrical system shown in the remainder of FIG. 1.
The electrical system portion of FIG. 1 shows an open loop fiber optical gyroscope system, but could also be converted to a closed loop fiber optic gyroscopic system, i.e. using feedback around the system shown. This would be accomplished by having the electrical system provide a feedback signal based on the output of the system shown in FIG. 1 to control a further optical phase modulator inserted in the optical path next to modulator 19, or to additionally control modulator 19. Optical phase modulator 19 is of the kind described above and is used in conjunction with a phase sensitive demodulator, or phase detector, for converting the output signal of photodiode 13 and photodetector system 14, following a cosine function, to a signal following a sine function. Following such a sine function provides, in that output signal, information both as to rate of rotation and as to direction of that rotation about the axis of coil 10. Modulator 19 is operated by a sinusoidal signal provided at the output of a bias modulation signal generator, 20, which also provides this signal to operate a phase detector which, as indicated, is a phase sensitive demodulator.
Thus, the output signal from photodetector system 14, including photodiode 13, is provided to an amplifier, 21, where it is amplified and passed through a filter, 22, to a phase detector, 23. The phase sensitive demodulator serving as phase detector 23 is a well-known device. Such a phase sensitive demodulator senses changes in the first harmonic, or fundamental frequency, of signal generator 20 to provide an indication of the relative phase of the pair of electromagnetic waves impinging on photodetector 13. This information is presented by phase detector 23 in an output signal following a sine function, i.e. the sine of the phase difference between the two electromagnetic wave portions impinging on photodiode 13.
Bias modulation signal generator 22, in modulating the electromagnetic wave portions in the optical path at a frequency set by the output signal supplied thereby as described above, also generates a strong second harmonic component in photodetector system 14. Filter 22 is a notch filter for removing this second harmonic component.
In operation, the phase difference changes in the two opposite direction propagating electromagnetic waves passing through coil 10 in the optical paths therethrough to reach photodiode 13 will lead to average net phase difference changes which will be relatively small, and which will vary relatively slowly compared to the phase difference changes due to optical phase modulator 19 and bias modulator signal generator 20. Any average phase difference shift due to the Sagnac effect will be merely shift the average phase difference between the electromagnetic waves, and the output signal from phase sensitive demodulator 23, after photodiode signal demodulation therein, will depend on the sine of this phase difference multiplied by an amplitude scaling factor set by the modulation of the waves due to phase modulator 19 and signal generator 20. This synchronous demodulation thus substantially extracts from the photodiode output signal the amplitude of the sinusoidal modulation frequency component at the modulation frequency introduced by signal generator 20 and modulator 19, which includes the result of any rotation of coil 10 about its axis, to provide the demodulator output signal.
As indicated above, however, additional phase shifts between the counter-propagating electromagnetic waves can be introduced even with the fiber optic gyroscope system in a minimum reciprocal configuration by various effects occurring therein. Typically, a significant source of such non-reciprocal phase shifts from other than the Sagnac effect is the following of different optical paths by the two different polarization components of the counter-propagating electromagnetic waves leading to phase shift errors in the output indistinguishable from Sagnac phase shifts.
Two types of such phase shift errors have been found to occur, as will be shown below. Amplitude type phase error occurs where electromagnetic waves that have passed through along the blocking axis of polarizer 15, because of polarizer imperfections, coherently mix in any of the loop optical components, beginning with loop coupler 17 and continuing beyond in FIG. 1 to include coil 10, with waves that have been passed along the transmission axis of polarizer 15. Since the electromagnetic waves emitted by source 11 along two linear polarization axes are uncorrelated, as will be set out below, and since these two polarization components are attempted to be kept separated until they reach polarizer 15, the conditions leading to amplitude type error will only occur through a failure to keep the components separated because of some coupling point being present ahead of polarizer 15 where such a polarization component can be partially coupled into the optical path of the other. Such coupling points include splice and interface locations in the optical paths because of the interruption introduced by such changes.
Intensity type phase error occurs when electromagnetic wave polarization components that have passed along the transmission axis of polarizer 15 are coupled in any of these same loop optical components to the polarization components which passed along the blocking axis, again due to polarizer imperfections, and thereafter reaches the blocking axis of polarizer 15 to interfere with the waves having the same history traveling in the opposite direction through coil 10. A phase shift error in the opposite direction occurs for waves doing the opposite, that is, passing along the blocking axis of polarizer 15 and being coupled to reach the transmission axis. Thus, these phase shift errors are offsetting of one another to the extent of the smaller of the two.
However, these unwanted phase shift errors are reduced or eliminated by the presence of depolarizer 10' in the system of FIG. 1 leading to the uniform (though not necessarily coherent) mixing of the electromagnetic wave components from the transmission and blocking axes of polarizer 15. That is, depolarizer 10' distributes portions of these incoming wave components into orthogonal polarization states such that they become thoroughly mixed at the other end of the depolarizer. The use of depolarizer 10' thus avoids the expense of using polarization-maintaining fiber throughout coil 10, but at the cost of losing in polarizer 15 approximately half of the electromagnetic wave energy entering coil 10.
All of these unwanted phase shifts would be eliminated, of course, if polarizer 15 were a perfect polarizing component permitting complete wave transmission in the transmission axis thereof and no transmission in the blocking axis thereof. However, polarizers, including polarizer 15 as indicated above, are not perfect and so are characterized by an extinction ratio, .epsilon., representing the fraction of incident electromagnetic wave on the polarizer in the blocking axis which is found in that same axis at the output of the polarizer. Such an imperfect polarizer, even with the presence of depolarizer 10', can lead to errors at photodiode 13 since coupling between polarization components at interfaces, splices and along the course of the ordinary single spatial mode optical fiber used in coil 10 results in effective non-reciprocal phase shifts.
Such errors in the transmission of electromagnetic waves in the system of FIG. 1 can be represented based on using a reference point between source coupler 12 and polarizer 15 in FIG. 1 since nothing fundamentally new or different will be added to these errors by the remaining propagation effects in reaching photodetector 13. This reference point will be represented as a "wavy" line in FIG. 1 at which outgoing and returning electromagnetic waves will be compared. Electromagnetic waves at this "wavy" line reference point will be represented as E.sub.x (t) and E.sub.y (t) for the components traveling along the principal birefringent axes of the polarization-maintaining optical fiber extending from this reference point to polarizer 15. The "x" designation, as indicated above, indicates the slow axis electromagnetic wave component and the "y" designation indicates the fast axis electromagnetic wave component.
The electromagnetic waves returning to the reference point after having left the reference point to traverse through coil 10 and the optical components therebetween can be written as follows: EQU .sub.cw (.nu.)=G.sub.cw (.nu.)e.sup.+j.phi.r .sub.i (.nu.) EQU .sub.ccw (.nu.)=G.sub.ccw (.nu.)e.sup.-j.phi.r .sub.i (.nu.)
where .nu. is the optical frequency. The vector .sub.i (.nu.) (designated as such through having an arrow over the top thereof) represents the electromagnetic waves leaving the reference plane to travel through coil 10 and return, and so is composed of just the two scalars E.sub.x (t) and E.sub.y (t) indicated above as occurring at the reference point but represented here by the Fourier transformation thereof, or ##EQU1## The transfer matrices representing the effects of the polarization-maintaining fiber, polarizer 15 and coupler 17, operating on the representation of the departing waves .sub.i from the reference point, yields the returning wave vector .sub.cw (.nu.) and .sub.cww (.nu.), and can be written as follows: ##EQU2## The Sagnac phase shift is represented by 2.phi..sub.r.
As described above, the output signal provided by phase sensitive demodulator 23 depends on the total phase shift, .DELTA..phi., occurring between the counter-propagating electromagnetic wave passing through coil 10 to reach photodetector 13. Thus, this output will depend essentially on the phase difference of the waves returning to the "wavy" line reference point which can be found from the argument of the complex matrix resulting from the product of the two waves, or EQU .DELTA..phi.(.nu.)=arg[ .sub.ccw.sup. (.nu.) .sub.cw (.nu.)]
The "dagger" symbol indicates that the Hermitian conjugate of the matrix is being used. This last equation can be rewritten using the equations above for .sub.cw (.nu.) and .sub.ccw (.nu.) and taking .phi..sub.r to be zero, i.e. no rotation of coil 10 about its axis perpendicular to the plane of the coil in FIG. 1, so only phase differences due to errors remain. Then, the phase differences, or path difference phase errors, .DELTA..phi..sub.e, due to polarization component path differences appear as: EQU .DELTA..phi..sub.e (.nu.)=arg[ .sup. .sub.i (.nu.)G.sup. .sub.ccw G.sub.cw .sub.i (.nu.)]
Here again, the indicates that the Hermitian conjugate of the matrix is being used. This last result is obtained at each optical frequency .nu., and so is the end result if a near monochromatic optical source is used for source 11. However, since source 11 is typically going to be a "broadband" optical source, the total error can be found only through integrating the last equation over the optical frequency although such a step is not made explicit here.
The off-diagonal elements of the transfer matrices [G.sub.cw ] and [G.sub.ccw ] in the system of FIG. 1 are much smaller in magnitude than the diagonal terms therein in an optical fiber based gyroscope using a polarizer despite the presence of depolarizer 10'. Since these off diagonal terms in these matrices are small, the last equation can be shown to be approximately: ##EQU3## That is, the total phase error .DELTA..phi..sub.e at any optical frequency .nu. can be separated into two parts, an amplitude related phase error, .DELTA..phi..sub.Ampl, and an intensity related phase error, .DELTA..phi..sub.Inten. The last two expressions show that amplitude related phase errors depend on the relative phases of the electromagnetic wave polarization components .sub.x (.nu.) and .sub.y (.nu.) and the first power of the polarization extinction coefficient, whereas the intensity related phase error depends on the differences in optical power in the two polarization component optical paths and the square of the polarizer extinction coefficient.
Clearly, errors can remain if .epsilon. has a value other than zero which it inevitably has. Thus, there is a desire for a measure or measures to reduce or eliminate such errors.